AS COMMONLY INVESTIGATED. 247 



M + N^h 



that enters the remainder, and thus introducing a new term — =-. The manner 



?h 



of obtaining the coefficients of such terms, and of thus extending the expansion 

 beyond the monoramic functions of the first order of continuity, must now oc- 

 cupy our attention. 



And on this point we may remark, that if in place of h" we use for our 

 classifying function any expression that fulfils the equation 



D. f h = (H) 



for all values of m that exceed n, and which is unity for m = n, we shall im- 



m 



mediately obtain the coefficient sought by performing the operation D on both 

 sides of the equation. 



This operation, J), is supposed to be of that class which makes the operation 

 performed, on the sum of any number of terms equal to the sum obtained, by 

 performing the operation on each separately; a condition conveniently ex- 

 pressed by the notation 



n 



Now suppose F x the function expanded ; F the same function diminished 



n 



by subtracting all the terms of the expansion up to the n*; and fx the classi- 

 fying function : in other words, let there be 



11 2 2 3 3 n n 



Fx = Pfx + P.fx + P.fx. . .Pfx. . . 



n n n 



F = P f X 4- &c. 



n 



and. it will follow, if f x fulfils the equation (11,) that 



n n n 



P = DF. 



In the instance before us the function to be expanded is F (x -f h;) and the 

 classifying function h", where m may be positive or negative, whole or frac- 

 tional; and all these cases will be included by supposing the values of n to 

 constitute the series 



." . . 1 



q" q" ^q"-'^ 



Now, supposing [i to be the least number which is divisible by all the deno- 

 minators, and denoting the quotients by (i', ii", &c., it is clear that an opera- 



n 



tion D which fulfils the equation (11) will consist in substituting k'" in place 



